= Solution
The <Gershgorin circle theorem> states that every <eigenvalue> of $A=(a_{ij})\in\mathbb R^{n\times n}$ belongs to at least one of the closed <Gershgorin discs>
$$
\boxed{D_i=\left\{z\in\mathbb C:|z-a_{ii}|\leq r_i\right\},\qquad r_i=\sum_{j\ne i}|a_{ij}|.}
$$
The <eigenvalue> may be complex even though the <matrix> is real. To prove the theorem, choose a nonzero complex <eigenvector> $v$ with $Av=\lambda v$ and an index $i$ at which $|v_i|$ is maximal. Then $|v_i|>0$, and the $i$th row gives
$$
(\lambda-a_{ii})v_i=\sum_{j\ne i}a_{ij}v_j.
$$
The <triangle inequality> yields
$$
|\lambda-a_{ii}|\leq\sum_{j\ne i}|a_{ij}|\frac{|v_j|}{|v_i|}\leq\sum_{j\ne i}|a_{ij}|=r_i.
$$
Thus $\lambda\in D_i$, proving the assertion without any symmetry assumption.
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